The Functional Thermodynamics of Finite-State Maxwellian Ratchets

Autonomous Maxwellian demons exploit structured environments as a resource to generate work: They randomize ordered inputs and leverage the increased Shannon entropy to transfer energy from a thermal reservoir to a work reservoir, respecting both Liouvillian state-space dynamics and the Second Law. To date, correctly determining such functional thermodynamic operating regimes was restricted to information engines for which correlations among information-bearing degrees of freedom can be calculated exactly via compact analytical forms---a highly restricted set of engines. Despite information-engine controllers being represented as finite hidden Markov chains and rate matrices, the restriction arose since the processes generated are notoriously complicated, defying closed-form analysis. Even when finite state (i) no finite expression for their Shannon entropy rate exists, (ii) the set of their predictive features is generically uncountably infinite, and (iii) their effective memory---the statistical complexity---diverges. When attempting to circumvent these challenges, previous analyses of thermodynamic function either heavily constrained controller structure, invoked approximations that misclassified thermodynamic functioning, or simply failed. Here, we adapt recent results from dynamical-system and ergodic theories that efficiently and accurately calculate the entropy rates and the rate of statistical complexity divergence of general hidden Markov processes. Applied to information ratchets, the results precisely determine the thermodynamic functionality for previously-studied Maxwellian Demons and open up analysis to a substantially broader class of information engines.